The University of Michigan Combinatorics Seminar
First I will discuss joint work with Alex Postnikov and David Speyer (arXiv:0706.2501) in which we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian G(k,n)+. G(k,n)+ is a cell complex whose cells C_G can be parameterized in terms of the combinatorics of certain planar graphs G. To each cell C_G we associate a certain polytope P(G), whose combinatorial structure is reminiscent of the well-known Birkhoff polytopes: the face lattice of P(G) can be described in terms of matchings and unions of matchings of G. We also demonstrate a connection between the polytopes P(G) and matroid polytopes. We then associate a toric variety to each P(G), and use our technology to prove that the cell decomposition of G(k,n)+ is in fact a CW complex.
Next I will briefly describe very recent joint work with Konstanze Rietsch in which we extend the previous work to show that the Lusztig-Rietsch cell decomposition of the totally non-negative part of any G/P is a CW complex. In the proof, the combinatorics of planar graphs is replaced by the technology of Lusztig's canonical basis.